Eksamen. Emnekode: Emnenavn: ME-408 Econometrics. Dato: Varighet 9:00-42:00. Antall sider inkl. forside 7. Tillatte hjelpemidler

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1 Eksamen Emnekode: Emnenavn: ME-408 Econometrics Dato: Varighet 9:00-42:00 Antall sider inkl. forside 7 Tillatte hjelpemidler Merknader -Dictionary (English-XX, XX-English) -Pocket calculator (non-programmable) -Collection of math formulae for economists You have to show all of your work. Be explicit! A maximum of 150 points can be obtained.

2 I Problem 1 [50pts] An accident study relates the dependent variable "accident rate" (per million vehicle miles) to 13 potential regressors. The observations available for the variables were measured on 39 sections of large highways in Minnesota in The statistician working on the problem initially formulates the "full" model as 13 -Yi i3o+ xii + Ei (1) j=1 with i = 1,, 39. She assumes that the standard assumptions of the classical linear model hold. Variable X2 X3 X4 X5 X6 X8 X9 X-10 X-11 X12 X-13 STATA variable RATE LEN ADT TRKS SLIM LWID SHLD ITG SIGS ACPT LANE FAI PA MA name Table 1: Accident data Interpretation accident rate per million vehicle miles length of segment in miles average daily traffic count iii thousands truck volume as percent of total volume speed limit lane width in feet width of out shoulder in feet number of freeway-type interchanges per mile number of signalized interchanges per mile number of access points per mile total number of lanes of traffic in both directions Federal aid interstate highway (=1;0 otherwise) principal arterial highway (=1; 0 otherwise) minor arterial highway (=1; 0 otherwise) The accident data were gathered on 4 types of highways: Major connectors, Federal interstates, principal highways and minor highways. Below, we give the coding scheme used in the study: Type of highway FAI PA MA Federal Principal Minor Major connectors

3 [a] Give the STATA code that triggers the regression output for the full model (5 pts) Source SS df MS Number of obs = 39 F( 13, 25) = 6.10 Model Prob > F Residual R-squared Adj R-squared = Total Root MSE RATE Coef. Std. Err. t P>ItI [95% Conf. Interval] LEN ADT TRKS SLIM LWID SHLD ITG SIGS ACPT LANE FAI PA MA _cons Interpret the estimates /312, and $13. Give the definition of the R2 measure and interpret the number given in the STATA output for the full model. (Hint: Before you judge the model on the basis of R2 consider the information concerning the significa,nce of the regressors given in the STATA output.) This item focuses on multicollinearity (MC). (10 pts) (10 pts) (15 pts) Define MC and explain how this phenomemon affects the estimation process. Define and explain the condition number K and the VITi. Based on the information given in Tables 2 and 3 discuss the relevance of multicollinearity in the context of the model at hand. Table 2: Eigenvalues of the correlation matrix for Xi,, X13 in (1) Al A2 A3 A4 A5 A6 A7 A8 A9 A10 All Al2 Al

4 Table 3: VIF's for the full model (1) Variable ADT FAI PA ITG MA SHLD SLIM LANE ACPT SIGS TRKS LEN LWID Mean VIF VIF [e] Consider the test of the hypothesis Ho 1313= 0 versus at least one = = /35 = 136= 7 = = (10 pts) Formulate this hypothesis as a general linear hypothesis Ho : Tr3 = 0 versus fl1 : T13 0, that is give the adequate matrix T. Give the STATA statement which will lead to the output given below. Test H0 based on the output given below and outline the consequences of your decision the process building a model for accident rates. STATA output: - LWID + LANE = 0 - ADT + LANE = 0 - SHLD + LANE = 0 LANE - FAI = 0 ITG + LANE = 0 LANE - MA = 0 - TRKS + LANE = 0 LANE - PA = 0 F(8,25) = 0.55 Prob > F =

5 1Problem2 [50pts] In this problem we study Canadian money demand for the period extending from 1956q1 to 1978q4 using a partial adjustment model. Quarterly data from Canada are available for the sample period 1956q1 to 1988q4. The data have been tsset. Consequently, when you are asked to write STATA statements, you can use time series statements/operators without further qualification. You have adhere to the notational conventions outlined in the table below. Table 1: Notational conventions Symbol Interpretation STATA variable name mt demand for real balances (M1) Yt real output at time t Rt long-term interest rate at time t trend variable trend t3, long-run parameters (i = 0, 1,2) short-run parameters (j = 0,... 4) The following long-run money demand equation is specified + /31yt + i32rt + vt (2) where vt is a,ssumed to be identically and independently distributed with N(0, av2). Consider the partial adjustment model (12 pts) mt mt-i = 0(mt* mt-1) (3) where m denotes the long-run real demand for money. Show that the regression model for mt mt = 'Yo+ YiYt + "Y2Rt + 2'3rnt- 1+ Et (4) can be deduced from (2) and (3). Identify the parameters "-Y2'-)'3 and the error term ft. To capture the persistent trends in the series at hand, we add a trend variable before (6 pts) estimating the model. mt = + 'YlYt+ "i2rt + 737flt-1 + -y4trend + et. (5) Give the STATA command code which will produce the output given below for the sample period 1956q1 to 1978q4. Source I SS df MS Number of obs = 91 + F( 4, 86) = Model Prob > F = Residual I R-squared = Adj R-squared =

6 Total I Root MSE = Coef. Std. Err. t P>ItI [95% Conf. Interval] R In Ll trend _cons Compute the estimates of the long-run parameters and [32 Interpret each param- (12 pts) eter. For the time period 1956q1 to 1978q4 the average income was = The average (10 pts) interest rate was observed to be = Compute the long-run elasticities of expected money demand with respect to income and interest and explain the meaning of your results. The following normal plot has been generated for the residuals from the regression (5) (10 pts) Empirical P[i] = i/(n+1) Figure 1: Normal plot for et, t E (1956q1, 1978q4) Give a short description of the rationale underlying the plot. What can you conclude about the validity of the normal assumption concerning the error vt? Explain. 6

7 Problem 3 [50 pts] Consider the multiple regression model = X (n,p),(p,1) + (n,1) with n > p. (25 pts) Suppose the errors are autocorrelated ft = pft_i +,yt with p i< 1. Derive the transformed model which provides the basis for the Cochrane-Orcutt procedure. Show that a distributed lag model (25 pts) Yt = L3o+ /310-Kt+ 310(1 0)Xt-1 + /310(1 0)2Xt_ (1 9)3Xt_3 + + ut with GE (0,1) can be represented as Yt = 'Yo+21.Xt + '72/7t-1 + ft with -yo= NO, = /310, = (1-0) and Et = Ut (1 0)ut. Give a short account of the practical implications of this result. 7